mapleprimes.com, as data

In essence, he discovered that the binary operator *EML*, along with the constant 1,forms a basis for the set of standard scientific-calculator operations.This means that functions like,and operations likeorcan be creating by composing*EML*with itself in clever ways. Some constants and functions are trivial to represent, such as*EML(1,1) = e*or*EML(x, 1) = exp(x)*, others however, are not...With a quick one-command tweak, you can get Maple to use the property of the extended reals thatAnd then with a simple argument about standard branches, you can construct the natural logarithm for real numbers, which immediately leads the constant zero:Which then expands the toolbox further by allowing for the construction unary minus from the constant 0 (since -x = 0 - x), and then addition (since a+b=a-(-b))Since we've constructed addition, subtraction, zero and one, we can technically construct every integer! It would not be very pleasant, and by no means optimal... but you could! Here's 7 for example:The next step to building all the standard functions is multiplication and inversion. And these use the classic trick by using the fact that x=exp(ln(x)) can help simplify:These are compositions of exp, addition, ln, and unary minus (all functions constructed previously), which means they can be made with only EML:It's at this point that I will leave the derivation of division (a/b) and exponentiation (a^b) as exercises for the reader, so I can skip to something a little more*complex...*To go beyond the basic operators, you'll need to step into the complex domain by constructing the imaginary constant i. To do this, take ln(-1) = -i*Pi (by using the standard branch) and combine it with Euler's formulaAnd once again the expression on the left-hand side is made up of operations that were all previously defined, so you can compose EML to get a new constant: